Introductory Mathematics and Statistics for Islamic Finance, + Website

Chapter 19Modeling Volatility: ARCH-GARCH Models

Modeling volatility is key for pricing assets and asset derivatives. Time-varying volatility of returns was initially studied in the context of an autoregressive conditional heteroskedasticity (ARCH) model. The ARCH model concentrates on the volatility dynamic. Financial markets display high volatility. Stock prices seem to go through a period of high volatility with significant changes in returns. Hence, volatility of returns may be a time-varying variable. If the variance of returns is time-varying then the risk variable has a time-varying variance. Many economic time series exhibit volatility clustering; that is, periods of unusually large volatility are followed by periods of relative tranquillity. Large changes in prices tend to be followed by large changes, of either sign, and small changes of prices tend to be followed by small changes. The assumption of linear effect of past shocks on returns is not realistic; likewise, the assumption of constant variance of returns is not supported by data.

We describe the ARCH model in this chapter. The term heteroskedasticity refers to changing volatility (i.e., variance) of returns. But it is not the variance itself that changes with time according to an ARCH model; rather, it is the conditional variance of returns that changes, in a specific way, depending on the available data. We analyze the properties of the ARCH model. The key insight offered by the ARCH model lies in the distinction ...